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1408 JOURNAL OF COMPUTERS, VOL. 8, NO. 6, JUNE 2013
new solution. State entry 9, 2, 3, and 7 from the best
solution form part of new solution. State entry 8, 3, 9, 2, 4
and 6 from current solution form the other part of new
solution. Modification of the new solution is carried out
to make sure that duplicated state entry 2, 3, and 9 do not
appear in the solution. By doing so, unassigned state
entry 0, 1 and 5 are filled in.
Figure 2. Exchange method.
Figure 3. Swap method for two states close to each other.
Figure 4. Swap method for two states apart from each other.
D. Crossover
Crossover operator is designed to escape the local
optimum in the GA. In this paper crossover operator is
adapted in the simulated annealing to generate a new
solution by combining the best solution with the current
solution. Consequently, some potential parts of the
solution can be inherited from the previous solution.
It operates as a position-based crossover [18]. An array
of binary bits is initialised randomly, in which the length
of the array equals to the length of the representation.
When 1s appear in the array, copy the states from the best
solution to new solution. When 0s appear in the array,
copy the states from the current solution to new solution.
Since each entry of state assignment solution is unique,
continuous check is required avoiding the repetition of
the same states which is not allowed. If state entry is
duplicated, the first unassigned state entry is assigned.
Figure 5 shows an example of relay operator to generate
Figure 5. Crossover operator.
E. Cost Function
By minimising (6), low power dissipation could be
achieved. Unfortunately, this may lead to area overhead,
resulting in power overhead in the combinational part of
the circuit. Therefore, objectives of area and power
should be optimised simultaneously. The total cost
function is C total = γC area + (1-γ) E, where C area is area
function, E is power function and γ is a parameter
specifying the tradeoff of E with respect to C area .
F. Proposed Algorithm
To begin with, an initiation solution is randomly
produced. The annealing approach is divided into rough
annealing and focusing annealing, which is identified by
Temperature threshold T threshold . In the rough annealing
stage the temperature is high and uses exchange method.
In the focusing annealing stage the temperature is low
and uses swap methods. The solution is evaluated by the
cost function. The solution is accepted according to a
probability P = exp (-∆C/T), where ∆C is the difference
cost between new solution and current solution, T is the
temperature. P is between 0 and 1. If the solution is
improved and kept according to P, the new solution will
be compared with the best recorded solution to decide
whether updating the best record is necessary. If new
solution is rejected instead, this solution is then ignored
and reverses the current solution. Outer loop updates the
temperature. The temperature schedules at each outer
iteration. In the inner loop it generates possible solutions
at each temperature. The algorithm terminates when
reaching the lowest temperature and the output will be the
best record. The pseudo code is shown in Algorithm 1.
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